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Fault feature extraction of rolling element bearings using sparse representation
Journal of Sound and Vibration 366 (2016) 514–527
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Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Fault feature extraction of rolling element bearings using sparse representation Guolin He, Kang Ding, Huibin Lin n School of Mechanical and Automotive Engineering, South China University of Technology, Guangzhou 510640, PR China
a r t i c l e in f o
abstract
Article history: Received 2 August 2015 Received in revised form 28 October 2015 Accepted 10 December 2015 Handling Editor: I. Trendafilova Available online 24 December 2015
Influenced by factors such as speed fluctuation, rolling element sliding and periodical variation of load distribution and impact force on the measuring direction of sensor, the impulse response signals caused by defective rolling bearing are non-stationary, and the amplitudes of the impulse may even drop to zero when the fault is out of load zone. The non-stationary characteristic and impulse missing phenomenon reduce the effectiveness of the commonly used demodulation method on rolling element bearing fault diagnosis. Based on sparse representation theories, a new approach for fault diagnosis of rolling element bearing is proposed. The over-complete dictionary is constructed by the unit impulse response function of damped second-order system, whose natural frequencies and relative damping ratios are directly identified from the fault signal by correlation filtering method. It leads to a high similarity between atoms and defect induced impulse, and also a sharply reduction of the redundancy of the dictionary. To improve the matching accuracy and calculation speed of sparse coefficient solving, the fault signal is divided into segments and the matching pursuit algorithm is carried out by segments. After splicing together all the reconstructed signals, the fault feature is extracted successfully. The simulation and experimental results show that the proposed method is effective for the fault diagnosis of rolling element bearing in large rolling element sliding and low signal to noise ratio circumstances. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Rolling element bearing Feature extraction Sparse representation Matching pursuit
1. Introduction Rolling element bearings are critical components of rotating machinery, and it’s important to monitor their condition to avoid catastrophic failures in modern machinery. A rolling element bearing usually consists of an inner race, an outer race, rollers and a cage. If a local damage develops on the surface of any of these components, the strikes of rollers on the fault surface will excite the resonant frequencies of structures between the bearing and the transducers, and trigger the periodical impulses which are characterized by modulation phenomenon [1,2]. The impulse response signal can be collected by transducer installed on the bearing pedestal accompanied by other structure vibration and noise. How to extract the impact signal accurately is a key issue of fault diagnosis of bearing. In the past twenty years, signal sparse representation theory [3] has received considerable attentions and made remarkable achievements in the field of image processing [4], speech recognition [5] and compressed sensing[6,7]. This
n
Corresponding author. Tel.: þ86 20 87113220. E-mail addresses: [email protected] (G. He), [email protected] (K. Ding), [email protected] (H. Lin).
http://dx.doi.org/10.1016/j.jsv.2015.12.020 0022-460X/& 2015 Elsevier Ltd. All rights reserved.
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theory is also be used for fault feature extraction and signal separation in rotating machinery recently. The foundation of the sparse representation is to construct signal based on a linear combination of basis functions or atoms. There are two key issues related to sparse representation: redundant dictionary design and sparse coefficients solving. In order to match the local structure of signal, the dictionary must be carefully designed; the more similar the dictionary’s atom and the structure of signal are, the more quickly and accurately the analyzed signal will be sparsely represented. For extracting the impact caused by bearing fault using the sparse decomposition, many works focused on how to construct dictionary which has similar structure with the analyzed signal. For example, Liu et al. [8] employed matching pursuit with time-frequency atoms to analyze bearing vibration and extract vibration signatures, and proved that the method worked better than continuous wavelet transform and envelope detection. Cui et al. [9] established a new impulse dictionary according to the characteristics of rolling bearing, and matching pursuit using the new dictionary combined with genetic algorithm was used for bearing fault diagnosis. Using the redundant Fourier basis, identity matrix basis and short-time Fourier basis, Qin et al. [10] constructed a composite transform basis dictionary and vibration signal components of faulty machine were separated by iteratively using basis pursuit algorithm. Zhu and Cai [11,12] employed Laplace wavelet, Morlet wavelet, tunable Q-factor wavelet successively to construct the redundancy dictionary, and fault features of rotating machines were extracted by the split augmented Lagrangian shrinkage algorithm (SALSA) combined with neural network. Tang et al. [13] employed shiftinvariant sparse coding (SISC) algorithm for dictionary learning, and the underlying structure of machinery fault signal was captured by sparse representation based on latent components decomposition method. Liu et al. [14] optimized atom parameters using spectrum kurtosis and smoothing index, and the faulty impulse period was matched by correlation matching method based on optimized atom. Although sparse representation has been used for diagnosing bearing fault successfully, there is still much work to do, and the following situations still need to be considered. (1) When a localized defect is induced, repeat impacts will be generated due to the passing of rolling elements over the defect. The wide-band energy of the impacts will evoke several modes of resonance of the bearing elements, the structure and the sensor. But most dictionaries constructed in literature only use one natural frequency. There is no guarantee that these dictionaries will match the structures of the analyzed data well. (2) For inner race and rolling element faults, the amplitudes of the defect-induced impulses vary as the inner race or rolling element defects enter and leave the bearing load zone [15,16]. Taking the transmission path and the projection direction of the impulse force into further consideration, the amplitudes of the impulse forces may be smaller or even harder to detect when the defect is out of the bearing load zone. For simplicity, most literature did not take the impulse missing into consideration in sparse coefficients solving. (3) The slippage of rolling element and the fluctuation of rotational speed may cause random variation in spacing between two consecutive defect-induced impulses in practice [17]. But most literature diagnosed the bearing fault by using sparse decomposition based on the consideration of equal-spaced generation of force impulses, and the periodically time-varying statistics characteristic of the vibration of rolling element bearing was neglected. To extend the suitability of sparse representation theory on bearing fault diagnosis, the above situations should be taken into consideration. In this paper, the natural frequencies and relative damping ratios of the bearing system are identified by correlation filtering method. The obtained waveform parameters are put into the impulse response function of damped second-order system to construct the redundant dictionary. Considering impulse missing and rolling elements sliding situations, the fault signal is divided into segments before sparse coefficient solving, and the defect-induced impulses are constructed by segments. After splicing together all the reconstructed segments, the averaging interval of the identified impulses is obtained and bearing fault can be diagnosed accordingly. The remainder of this paper is organized as follows: In Section 2, the principles of matching pursuit is briefly introduced. In Section 3, dictionary construction method based on correlation filtering is given, and a new bearing fault detection method using sparse representation with the designed dictionary is proposed. The effectiveness of the proposed method is verified by simulation in Section 4 and assessed by experimental results in Section 5. Finally, conclusions are drawn in Section 6.
2. The basic principles of matching pursuit Matching pursuit is a commonly used algorithm for sparse representation [18], and the principle of matching pursuit lies in decomposing a signal x into the linear combination of basis functions dγ (‖dγ ‖ ¼ 1) that belong to a redundant dictionary D A Rnq . After greedy search, an atom dγ 0 A D that best matches the signal structure is selected, and the signal x can be decomposed into (1) x ¼ x; dγ 0 dγ 0 þ R1 x where hi means inner product, R1 x is the residue after the first matching. For the orthogonality of dγ 0 and R1 x, we get ‖x‖2 ¼ x; dγ 0 2 þ‖R1 x‖2 (2)
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where ‖‖ means 2-norm. The residue R1 x is further decomposed in the same way as x. After applying N times similar procedure to the residue, dγ n is selected if Rn x; dγ n is the maximum over the dictionary, then x¼
N 1 X
Rn x; dγ n dγ n þ RN x
(3)
n¼0
This procedure is repeated until a desired approximated precision is reached (the final residue RN x is small enough). The matching pursuit approach works well only when the dictionary has the similar structure with the signal. In practice, however, it is difficult to find a robust parametric basis function that achieves goals of the feature extraction, which restricts the application of matching pursuit seriously.
3. The extraction of defect-induced impulse based on sparse decomposition 3.1. Construct the sparse dictionary based on correlation filtering When a localized fault occurs in rolling element bearing, the fault impact will produce structure resonance and generate impulse response signal with an exponential damping amplitude [19]. Consequently, damped second-order mass-springdamper system and its unit-impulse response function can be used to describe the vibration response of bearing fault, and the basis function of the sparse dictionary can be defined as 2 3 6 2πζ 7 dγ ðtÞ ¼ exp4qffiffiffiffiffiffiffiffiffiffiffiffiffif d ðt t 0 Þ5 sin 2π f d ðt t 0 Þ ; t Z t 0 (4) 2 1ζ where f d is the natural frequency of the bearing system. ζ represents the relative damping ratio which is generally less than 0.2 for steal structure [20]. t 0 is the moment that impact occurred. Obviously, the parameters ðf d ; ζ ; t 0 Þ, which decide the waveform of atoms, have an influence on the result of sparse decomposition. In the following, the parameters ðf d ; ζ ; t 0 Þ are optimized by correlation filter method [11] to approximate those impulses in the bearing signal xðtÞ. The followings are the procedure: (1) Let f d A f lc : Δf d : f s =2 , ζ A 0:001: Δζ : 0:2 and t 0 A 0: Δt: T c , where Δfd , Δζ and Δt are the step-sizes of ðf d ; ζ ; t 0 Þ, f lc is the lower cutoff frequency, f s is sampling frequency, and T c is the signal duration that used for correlation filtering. The initial dictionary C is generated according to Eq. (4). For the periodic characteristic of the impulse response, only a short time duration T c which includes several impacts is needed. (2) Calculate the correlation coefficients λðtÞ between the bearing signal xðtÞ and the basis atoms dγ in the initial dictionary C. xðtÞ; dγ ðtÞ λðtÞ ¼ (5) ‖xðtÞ‖2 ‖dγ ðtÞ‖2 (3) Search all the local maximums λi ði ¼ 1; …NÞ from λðtÞ, and the corresponding natural frequencies, relative damping ratios and the occurrence times of impulses are denoted as ðf di ; ζ i ; t 0i Þ, i ¼ 1; …N. The search precision of above three parameters is directly influenced by the step-sizes Δfd, Δζ, Δt. (In practice, bearing fault may evoke several degrees of natural frequencies, meanwhile, considering the quantization error, the identified f di and ζ i of different local maximums λi may be same or close.) (4) Merge the identified natural frequencies close to each other into their middle value. In this paper, the area is set to f di 2Δf d ; f di þ 2Δf d . Let f dj ðj ¼ 1; …JÞ be the natural frequencies after merged, and ζ j be the corresponding relative damping ratios. (5) Put all the f dj ; ζ j ðj ¼ 1; …JÞ and t 0i ði ¼ 1; :::NÞ into Eq. (4), and get the optimized dictionary D with J N atoms. Obviously the redundancy of dictionary D is far less than that of initial dictionary C, which can largely improve the speed of sparse decomposition. Since the parameters of dictionary D are determined from the bearing signal xðtÞ, all the parameters have clear physical meaning. Moreover, several natural frequencies and damping ratios are used for constructing the impact response, which is in conformity with practical situations. 3.2. The proposed fault feature extraction method based on sparse decomposition To improve the calculation speed of sparse coefficients solving and make sparse representation more suitable for bearing fault diagnosis, the proposed method employs correlation filtering to get the optimized dictionary parameters based on the bearing signal firstly. Then the bearing signal is divided into several segments based on the smallest fault characteristic period, and the matching pursuit is carried out by segments. Moreover, to improve the extracting accuracy of the impact
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Fig. 1. Flowchart of the proposed fault feature extraction algorithm for rolling element bearing.
moment, the parameters of t 0i in the dictionary is further refined when executing matching pursuit. The process of the proposed algorithm is shown in Fig. 1. The details of this algorithm are summarized as follows: (1) Collect the bearing signal xðtÞ by acceleration transducer. (2) High pass filtering is carried on to remove the low frequency components caused by manufacturing and assembling errors and other structure vibration. Let xp ðtÞ be the signal after filtering. (3) Repeat the steps of Section 3.1, the dictionary parameters f dj ; ζ j ðj ¼ 1; …JÞ and t 0i ði ¼ 1; …NÞ are obtained from the signal xp ðtÞ by correlation filtering. (4) Let T s be the smallest appearance period of possible defect-induced impulse (which is corresponding to the inner race fault characteristic period). Divide the signal xp ðtÞ into W segments xw ðtÞðw ¼ 1; 2; ⋯; WÞ with each segment having a length of T s . For the single fault situation, this step can avoid more than one impact happened during one period. (5) Matching pursuit is applied for every segment xw ðtÞ to extract the impulse signal xws . Since we divide signal into segments by the characteristic period of inner race fault, there may be no impulse in some segments when local defect appears in other bearing parts. Even for inner race fault, the defect-induced impulse might be too weak to be detected when the defect location rotates out of the bearing load zone. In view of above situation, we utilize the parameter t 0i obtained from the correlation filtering to prevent the mismatch problem, and the details are as follows: A) Perform the division operation ðt 0i =T s Þ and seek its quotient and remainder, that is, ki ¼ roundðt 0i =T s Þ, τi ¼ modðt 0i ; T s Þ. B) Compare ki with w, when w aki , we can assume that there is no impulse or the impulse is too weak to be detected in the segment xw ðtÞ. In this case, the matching pursuit is not needed, and the reconstructed signal of this segment is set to zero. C) For segment xw ðtÞ that w ¼ ki , extend the remainder τi by the step-size Δτ to 2M þ 1 points, that is,τ0i ¼ ½ M Δτ þ τi ; M Δτ þ τi . Get the optimized dictionary Di by substituting f dj ,ζ j ðj ¼ 1; …JÞ and τ0i into the Eq. (4), and the atoms number of Di is J ð2M þ1Þ. Matching pursuit is applied to segment xw ðtÞ until the iteration v termination condition is reached. Let the best matching atom be dγ and corresponding matching coefficient cwv ¼
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D E v Rwv xw ; dγ ðv ¼ 0; :::V 1Þ, which are obtained at the end of the sparse decomposition. The reconstructed signal xws is xws ¼
VX 1 v¼0
v
cwv dγ
(6)
(6) The W segments of the reconstructed signals are sequentially connected and the impulse component of the bearing signal is obtained by xs ðtÞ ¼
W X
xws ½t þ ðw 1ÞT s
(7)
w¼1
(7) Get all the time interval of impulses in xs ðtÞ and take the average after discarding the value that changes largely, and obtain the fault feature finally.
4. Simulation analysis Considering the fact that several degrees of natural frequencies may be evoked by a local fault, the single point defect signal of rolling element bearing x(t) can be described through extending the bearing model proposed by Mcfadden [19] to
Fig. 2. Simulation signals of outer race fault with no noise. (a) Signal without slipping. (b) Signal with large slipping. (c) Amplitude spectrum of (a). (d) Amplitude spectrum of (b). (e) Hilbert envelop spectrum of (a). (f) Hilbert envelop spectrum of (b).
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Fig. 3. The simulated noisy signals (red line represents pure signal; blue line represents noisy signal). (a) SNR ¼ 0 dB. (b) SNR¼ 5 dB. (c) SNR¼ 10 dB. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 4. The process of correlation filtering for the pure signal with no slipping.
the following form 2
3 h i 6 2πζ j 7 Aij exp4qffiffiffiffiffiffiffiffiffiffiffiffiffif dj ðt τi iT Þ5 sin 2π f dj ðt τi iT Þ þ ηðt Þ x ðt Þ ¼ 2 i¼1j¼1 1 ζj J I X X
t Z τi
(8)
where ðf dj ; ζ j Þ j ¼ 1; …; J represent the jth natural frequency and corresponding relative damping ratio. Aij is the amplitude of the ith transient response under the jth natural frequency. T is the time period of the defect-related impulses, τi is the delay which is a random variable for the time lag between impacts due to the presence of sliding. ηðt Þ is additive background noise. Obviously, τi ¼ 0 when there is no sliding between the rolling elements.
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Fig. 5. (a) The identified results of natural frequencies and relative damping ratios for the pure signal with no sliding. (b) The identified natural frequencies for the case of large sliding. (c) The identified relative damping ratios for the case of large sliding.
Let the simulated bearing has the same parameters as bearing NUP311EN (pitch diameter D ¼ 85 mm, ball diameter d ¼ 18 mm, ball number z ¼ 13, contact angle α ¼ 0o ). Let f n be the shaft rotational frequency, and the ball passing inner race frequency (BPFI) f I and ball passing outer race frequency (BPFO) f O can be calculated by [9]
z fn d 1 cos α (9) fO ¼ D 2 fI ¼
z fn d 1 þ cos α D 2
(10)
When f n ¼ 10 Hz, the characteristic defect frequencies of bearing NUP311EN are: f O ¼ 51:24 Hz and f I ¼ 78:76 Hz. Let J ¼ 2, f d1 ¼ 1800 Hz, ζ 1 ¼ 0:03, f d2 ¼ 4200 Hz, ζ 2 ¼ 0:01, Aij is arbitrary constant distributed in the range of 0–10. Let the sampling frequency f s be 214 ¼16384 Hz, and sampling time is set to be 0.5 s. Let T ¼ 1=f O ¼ 19:52 ms in Eq. (8) for simulating outer race fault. Let τi ¼ 0 to simulate the case without sliding of rolling element and τi ¼ 0:5T randð0; 1Þ to simulate large sliding of rolling element, and the simulation signals are given in Fig. 2. By comparing Fig. 2(a) with (b), it can be seen that because of the influence of rolling element sliding, the time intervals between any two impulses are no longer fixed value, and serious frequency aliasing happens in Fig. 2(d). The two signals are both demodulated in the same bandwidth [1600, 2000] Hz and the envelop spectrums are given in Fig. 2(e) and (f), respectively. It can be seen that the fault characteristic frequency f O and its harmonics are no longer obvious in the demodulation spectrum for the large sliding of rolling element.
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Fig. 6. The reconstructed impulses under different SNR (red lines: reconstructed signal, blue line: theoretical signal). (a) Pure signal. (b) SNR ¼0 dB. (c) SNR ¼ 5 dB. (d) SNR¼ 10 dB. Table 1 Impulse interval of the reconstructed signals under different SNRs.
Average (ms) Relative error
set vale
pure signal
0 dB
5 dB
10 dB
19.52 /
19.45 0.36%
19.45 0.36%
19.43 0.46%
19.15 1.90%
A white gaussian noise is added to the simulated signal with large rolling element sliding. Let the signal to noise ratio (SNR) be 0 dB, 5 dB and 10 dB, respectively, and the noisy signals are given in Fig. 3. As can be seen, with the increasement of noise level, the fault impacts are buried in the noise gradually and become hard to detect at last. The above simulation signals are decomposed and reconstructed by the proposed fault feature extraction method. When searching the parameters by correlation filtering, the search scopes of f d ,ζ and t 0 are set as: f d A ½0: 10: f s =2, ζ A ½0:001: 0:001: 0:2 and t 0 ¼ ½0: T=24 : 0:5, respectively. The process of correlation filtering for the case τi ¼ 0 and ηðtÞ ¼ 0 is given in Fig. 4. The identified natural frequencies and corresponding relative damping ratios in different SNRs are given in Fig. 5 (where the identified natural frequencies are sorted from small to large before displaying). As is shown Fig. 5, both the two parameters are accurately identified when τi ¼ 0 and ηðtÞ ¼ 0, the identified accuracy of the natural frequency f di still keep high even under large sliding of rolling element and low SNR, and although the identified accuracy of relative damping ratios ζ i is degraded with the increasement of noise level, most the identified results distribute near the theoretical values. Considering the fact that only the most matching atoms are selected for constructing signal in matching pursuit, a good matching result can be obtained even when only one set of identified atom parameters is close to the real values. Consequently, all the identified parameters are used to construct the redundant dictionary. After dividing the simulation signal xðtÞ by the smallest appearance period T s of possible defect-induced impact (here T s ¼ 1=f I ¼ 1=78:76 ¼ 12:7 ms), matching pursuit is carried out on every segment xw ðtÞ following the step (5) in Section 3.2, where step-size Δτ is set as T I =27 and M is set as 10 when constructing the dictionary Di. The defect-induced impulses are reconstructed and the results are presented in Fig. 6. It can be seen that the reconstructed results match well with the theoretical results although there is a large rolling element sliding. And as shown in Fig. 6(b), the matching accuracy remains good when SNR ¼0 dB, but when SNR decreases to 10 dB, as shown in Fig. 6(d), some undesired signals are wrongly extracted. After discarding the suddenly changed value, the average and relative error of the impulse intervals are given in Table 1. It could be concluded that although the matching accuracy degrades with the decrease of SNR, the average of impact intervals under low SNR is still close to the set value. The simulation results show that the proposed method can extract the bearing fault feature even under large rolling element sliding and low SNR conditions.
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Fig. 7. Experimental facility: (a) The test bench; (b) The artificial fault in outer race.
Fig. 8. The vibration signal picked up from the test bench. (a) Time-domain wave. (b) Amplitude spectrum.
Fig. 9. The signal after high pass filtering xp(t).
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Fig. 10. The identified natural frequencies and relative damping ratios by correlation filtering. Table 2 The natural frequencies and relative damping ratios for optimized dictionary. j
1
2
3
4
5
6
7
8
f dj =ðHzÞ
7550
8000
9350
14750
15100
16150
16900
35750
ζj
0.021
0.026
0.091
0.055
0.050
0.026
0.020
0.011
Fig. 11. The reconstructed impulses for bearing with outer race fault.
Fig. 12. Hilbert envelop spectrum of the reconstructed bearing signal.
5. Experimental analysis 5.1. outer race fault experiment The experiment is conducted on a test bench shown in Fig. 7, the shaft of the test rig is supported by two bearings model N205M (pitch diameter D ¼38 mm, ball diameter d¼6.5 mm, ball number z¼13, contact angle α ¼ 0o ) and driven by an AC motor and a belt pulley. An artificial fault (width 0.2 mm and depth 0.5 mm) is produced on the outer-race of one bearing, and the vibration signal is picked up from the bearing pedestal with sample frequency fs ¼100 kHz. Let shaft rotational speed be 800 rpm, the characteristic defect frequencies (periods) of outer and inner races are fO ¼71.84 Hz (13.92ms) and fI ¼101.5 Hz (9.85ms), respectively. The time domain wave and frequency spectrum of the measured acceleration signal is presented in Fig. 8. The fault signal x(t) is decomposed and reconstructed according to the flowchart presented in Fig. 1. Let the lower cutoff frequency f lc ¼ 1=20f s to remove the low frequency disturbance caused by motor and pulley drive, and the filtered signal xp ðtÞ is shown in Fig. (9). The filtered signal is divided into equal segments based on the smallest fault characteristic period T s (9.85ms). Let f d A ½f lc : 50: f s =2, ζ A ½0:001: 0:001: 0:2 and t 0 ¼ ½0: T=24 : 1, when correlation filtering is carried out on the filtered signal xp ðtÞ. The identified natural frequencies f di and corresponding relative damping ratios ζ i are given in Fig. 10. After merging the adjacent values of f di , the parameters of optimized dictionary are shown in Table 2, and the final reconstructed signal is given in Fig. 11. By comparing Fig. 9 with Fig. 11, it can be seen that most interference components are removed in the reconstructed signal. After discarding the intervals that change largely, the average of the impact intervals in the reconstructed signal is 14.00 ms, which is very close to the characteristic defect period of outer race (13.92 ms). The
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Fig. 13. The transmission test bench and the fault bearing. (a) The tested transmission. (b) The position of sensor. (c) The artificial fault in inner race.
Fig. 14. The vibration signal picked up from the transmission. (a) Time-domain wave. (b) Amplitude spectrum.
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Fig. 15. The gearbox signal after high pass filtering xp(t).
Fig. 16. The identified results of natural frequencies and relative damping ratios of the gearbox signal. Table 3 The parameters of optimized dictionary. j
1
2
3
4
5
6
7
8
f dj = ðHzÞ
5450
6000
7650
12200
16200
17050
17850
45750
ζj
0.039
0.044
0.031
0.016
0.004
0.006
0.003
0.006
Fig. 17. The reconstructed impulses of the gearbox signal.
reconstructed signal is further demodulated in full frequency band, and the Hilbert envelope spectrum is shown in Fig. 12, in which the characteristic defect frequency of outer race f O and its harmonics are clearly exposed. Therefore, the outer race defect is identified. 5.2. Inner-race fault experiment To further verify the effectiveness of the proposed method, experiment is conducted on a five-speed automobile transmission. An artificial fault (width 0.2 mm and depth 1 mm) is produced on the inner-race of the bearing NUP311EN which supports the output shaft. The transmission test bench and the fault bearing are shown in Fig. 13. The rotational speed of the output shaft is set as 500 rpm, the characteristic defect frequencies (periods) of outer and inner races are computed by Eqs. (9) and (10), and the values are fO ¼42.75 Hz (23.39 ms) and fI ¼65.53 Hz (15.22 ms), respectively. Let the sample frequency fs ¼ 100 kHz, the vibration signal picked up from the bearing pedestal of output shaft is shown in Fig. 14. It can be seen that there is no visible impact in the time domain wave, and the amplitude spectrum is dominated by constantly mesh frequency and its harmonics. The signal is processed by the proposed fault feature extraction method with the same process parameters as Section 5.1. The signal after filtering is given in Fig. 15, where no clear fault impulse can be observed. Again, after the correlation filtering, the dictionary parameters are identified and the results are given in Fig. 16. By utilizing the optimized natural frequencies and relative damping ratios shown in Table 3, the fault impulses are reconstructed by segments and the final results are given in Fig. 17. As can be seen in Fig. 17, the periodicity of the impulses experiences a certain degree of randomness, and the cyclic statistics characteristics of the bearing are revealed clearly. This
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Fig. 18. Hilbert envelop spectrum of the reconstructed gearbox signal.
phenomenon is caused by the slippage as the rolling elements enter and leave the bearing load zone. After discarding the intervals that change largely, the average of the impact intervals is 15.47 ms. Comparing with the characteristic defect periods of bearing NUP311EN, there is only 1.6% relative error with the characteristics defect period of inner race (15.22 ms). Similarly, the reconstructed signal is further demodulated in full frequency band, and the Hilbert envelope spectrum is shown in Fig. 18. As can be seen, not only the characteristic defect frequency of inner race f I and its harmonics but also their sidebands are clearly detected. Thus we can confirm that fault should locate in the inner race of the bearing NUP311EN. The effectiveness of the proposed method is further verified.
6. Conclusions This paper proposes a bearing fault feature extraction method based on sparse representation. According to the characteristic of the defect-induced impulse, sparse dictionary is constructed by unit-impulse response function of damped second-order system, and the natural frequencies and relative damping ratios are obtained directly from bearing signal by correlation filtering method. Considering the influences of speed fluctuation, rolling element sliding and other time-varying factors, the defect induced impulses are reconstructed by segments. The results of both simulated and actual bearing vibration signals show that the proposed method is effective to extract the rolling bearing fault impulses even under large rolling element sliding and low signal to noise ratio.
Acknowledgments This work was supported by the National Natural Science Foundation of China (Nos. 51475169 and 51475170) and the Natural Science Foundation of Guangdong Province (S2013030013355).
References [1] Peter W. Tse, D. Wang, The design of a new sparsogram for fast bearing fault diagnosis: Part 1 of the two related manuscripts that have a joint title as “Two automatic vibration-based fault diagnostic methods using the novel sparsity measurement – Parts 1 and 2”, Mechanical Systems and Signal Processing 40 (2013) 499–519. [2] Peter W. Tse, D. Wang, The automatic selection of an optimal wavelet filter and its enhancement by the new sparsogram for bearing fault detection: Part 2 of the two related manuscripts that have a joint title as “Two automatic vibration-based fault diagnostic methods using the novel sparsity measurement-Parts 1 and 2”, Mechanical Systems and Signal Processing 40 (2013) 520–544. [3] S. Mallat, Z. Zhang, Matching pursuits with time-frequency dictionaries, IEEE Transactions on Signal Processing 41 (12) (1993) 3397–3415. [4] M.N. Do, M. Vetterli, The contourlet transform: an efficient directional multiresolution image representation, IEEE Transactions on Image Processing 14 (12) (2005) 2091–2106. [5] J.F. Gemmeke, T. Virtanen., A. Hurmalainen, Exemplar-based sparse representations for noise robust automatic speech recognition, IEEE Transactions on Audio, Speech and Language Processing 19 (7) (2011) 2067–2080. [6] M. Lustig, D.L. Donoho, J.M. Pauly, M.R.I. Sparse, the application of compressed sensing for rapid MR imaging, Magnetic Resonance in Medicine 58 (6) (2007) 1182–1195. [7] X.F. Chen, Z.H. Du, J.M. Li, X. Li, H. Zhang, Compressed sensing based on dictionary learning for extracting impulse components, Signal Processing 96 (2014) 94–109. [8] B. Liu, S.F. Ling, R. Gribonval, Bearing failure detection using matching pursuit, NDTE International 35 (2002) 255–262. [9] L.L. Cui, J. Wang, S. Lee, Matching pursuit of an adaptive impulse dictionary for bearing fault diagnosis, Journal of Sound and Vibration 333 (2014) 2840–2862. [10] Y. Qin, Y.F. Mao, B.P. Tang, Vibration signal component separation by iteratively using basis pursuit and its application in mechanical fault detection, Journal of Sound and Vibration 332 (2013) 5217–5235. [11] S. Wang, W. Huang, Z.K. Zhu, Transient modeling and parameter identification based on wavelet and correlation filtering for rotating machine fault diagnosis, Mechanical Systems and Signal Processing 25 (4) (2011) 1299–1320. [12] S.B. Wang, G.G. Cai, Z.K. Zhu, W.G. Huang, X.W. Zhang, Transient signal analysis based on Levenberg- Marquardt method for fault feature extraction of rotating machines, Mechanical Systems and Signal Processing 54-55 (2015) 16–40. [13] H.F. Tang, J. Chen, G.M. Dong, Sparse representation based latent components analysis for machinery weak fault detection, Mechanical Systems and Signal Processing 46 (2014) 373–388. [14] X.F. Liu, L. Bo, X. He, M. Veidt, Application of correlation matching for automatic bearing fault diagnosis, Journal of Sound and Vibration 331 (2012) 5838–5852.
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[15] N. Tandon, A. Choudhury, An analytical model for the prediction of the vibration response of rolling element bearings due to a localized defect, Journal of Sound and Vibration 205 (3) (1997) 275–292. [16] J. Antoni, R.B. Randall, A stochastic model for simulation and diagnostics of rolling element bearings with localized faults, Journal of Vibration and Acoustics 125 (3) (2003) 282–289. [17] R.B. Randall, J. Antoni, S. Chobsaard, The relationship between spectral correlation and envelope analysis in the diagnostics of bearing faults and other cyclostationary machine signals, Mechanical Systems and Signal Processing 15 (5) (2001) 945–962. [18] H. Fan, Q.F. Meng, Y.Y. Zhang, Q. Gao, F. Wang, Matching pursuit based on nonparametric waveform estimation, Digital Signal Processing 19 (2009) 583–595. [19] P.D. McFadden, J.D. Smith, Model for the vibration produced by a single point defect in a rolling element bearing, Journal of Sound and Vibration 96 (1983) 69–82. [20] J.D. Stevenson, Structural damping values as a function of dynamic response stress and deformation levels, Nuclear Engineering and Design 60 (1980) 211–237.
Contents lists available at ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Fault feature extraction of rolling element bearings using sparse representation Guolin He, Kang Ding, Huibin Lin n School of Mechanical and Automotive Engineering, South China University of Technology, Guangzhou 510640, PR China
a r t i c l e in f o
abstract
Article history: Received 2 August 2015 Received in revised form 28 October 2015 Accepted 10 December 2015 Handling Editor: I. Trendafilova Available online 24 December 2015
Influenced by factors such as speed fluctuation, rolling element sliding and periodical variation of load distribution and impact force on the measuring direction of sensor, the impulse response signals caused by defective rolling bearing are non-stationary, and the amplitudes of the impulse may even drop to zero when the fault is out of load zone. The non-stationary characteristic and impulse missing phenomenon reduce the effectiveness of the commonly used demodulation method on rolling element bearing fault diagnosis. Based on sparse representation theories, a new approach for fault diagnosis of rolling element bearing is proposed. The over-complete dictionary is constructed by the unit impulse response function of damped second-order system, whose natural frequencies and relative damping ratios are directly identified from the fault signal by correlation filtering method. It leads to a high similarity between atoms and defect induced impulse, and also a sharply reduction of the redundancy of the dictionary. To improve the matching accuracy and calculation speed of sparse coefficient solving, the fault signal is divided into segments and the matching pursuit algorithm is carried out by segments. After splicing together all the reconstructed signals, the fault feature is extracted successfully. The simulation and experimental results show that the proposed method is effective for the fault diagnosis of rolling element bearing in large rolling element sliding and low signal to noise ratio circumstances. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Rolling element bearing Feature extraction Sparse representation Matching pursuit
1. Introduction Rolling element bearings are critical components of rotating machinery, and it’s important to monitor their condition to avoid catastrophic failures in modern machinery. A rolling element bearing usually consists of an inner race, an outer race, rollers and a cage. If a local damage develops on the surface of any of these components, the strikes of rollers on the fault surface will excite the resonant frequencies of structures between the bearing and the transducers, and trigger the periodical impulses which are characterized by modulation phenomenon [1,2]. The impulse response signal can be collected by transducer installed on the bearing pedestal accompanied by other structure vibration and noise. How to extract the impact signal accurately is a key issue of fault diagnosis of bearing. In the past twenty years, signal sparse representation theory [3] has received considerable attentions and made remarkable achievements in the field of image processing [4], speech recognition [5] and compressed sensing[6,7]. This
n
Corresponding author. Tel.: þ86 20 87113220. E-mail addresses: [email protected] (G. He), [email protected] (K. Ding), [email protected] (H. Lin).
http://dx.doi.org/10.1016/j.jsv.2015.12.020 0022-460X/& 2015 Elsevier Ltd. All rights reserved.
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theory is also be used for fault feature extraction and signal separation in rotating machinery recently. The foundation of the sparse representation is to construct signal based on a linear combination of basis functions or atoms. There are two key issues related to sparse representation: redundant dictionary design and sparse coefficients solving. In order to match the local structure of signal, the dictionary must be carefully designed; the more similar the dictionary’s atom and the structure of signal are, the more quickly and accurately the analyzed signal will be sparsely represented. For extracting the impact caused by bearing fault using the sparse decomposition, many works focused on how to construct dictionary which has similar structure with the analyzed signal. For example, Liu et al. [8] employed matching pursuit with time-frequency atoms to analyze bearing vibration and extract vibration signatures, and proved that the method worked better than continuous wavelet transform and envelope detection. Cui et al. [9] established a new impulse dictionary according to the characteristics of rolling bearing, and matching pursuit using the new dictionary combined with genetic algorithm was used for bearing fault diagnosis. Using the redundant Fourier basis, identity matrix basis and short-time Fourier basis, Qin et al. [10] constructed a composite transform basis dictionary and vibration signal components of faulty machine were separated by iteratively using basis pursuit algorithm. Zhu and Cai [11,12] employed Laplace wavelet, Morlet wavelet, tunable Q-factor wavelet successively to construct the redundancy dictionary, and fault features of rotating machines were extracted by the split augmented Lagrangian shrinkage algorithm (SALSA) combined with neural network. Tang et al. [13] employed shiftinvariant sparse coding (SISC) algorithm for dictionary learning, and the underlying structure of machinery fault signal was captured by sparse representation based on latent components decomposition method. Liu et al. [14] optimized atom parameters using spectrum kurtosis and smoothing index, and the faulty impulse period was matched by correlation matching method based on optimized atom. Although sparse representation has been used for diagnosing bearing fault successfully, there is still much work to do, and the following situations still need to be considered. (1) When a localized defect is induced, repeat impacts will be generated due to the passing of rolling elements over the defect. The wide-band energy of the impacts will evoke several modes of resonance of the bearing elements, the structure and the sensor. But most dictionaries constructed in literature only use one natural frequency. There is no guarantee that these dictionaries will match the structures of the analyzed data well. (2) For inner race and rolling element faults, the amplitudes of the defect-induced impulses vary as the inner race or rolling element defects enter and leave the bearing load zone [15,16]. Taking the transmission path and the projection direction of the impulse force into further consideration, the amplitudes of the impulse forces may be smaller or even harder to detect when the defect is out of the bearing load zone. For simplicity, most literature did not take the impulse missing into consideration in sparse coefficients solving. (3) The slippage of rolling element and the fluctuation of rotational speed may cause random variation in spacing between two consecutive defect-induced impulses in practice [17]. But most literature diagnosed the bearing fault by using sparse decomposition based on the consideration of equal-spaced generation of force impulses, and the periodically time-varying statistics characteristic of the vibration of rolling element bearing was neglected. To extend the suitability of sparse representation theory on bearing fault diagnosis, the above situations should be taken into consideration. In this paper, the natural frequencies and relative damping ratios of the bearing system are identified by correlation filtering method. The obtained waveform parameters are put into the impulse response function of damped second-order system to construct the redundant dictionary. Considering impulse missing and rolling elements sliding situations, the fault signal is divided into segments before sparse coefficient solving, and the defect-induced impulses are constructed by segments. After splicing together all the reconstructed segments, the averaging interval of the identified impulses is obtained and bearing fault can be diagnosed accordingly. The remainder of this paper is organized as follows: In Section 2, the principles of matching pursuit is briefly introduced. In Section 3, dictionary construction method based on correlation filtering is given, and a new bearing fault detection method using sparse representation with the designed dictionary is proposed. The effectiveness of the proposed method is verified by simulation in Section 4 and assessed by experimental results in Section 5. Finally, conclusions are drawn in Section 6.
2. The basic principles of matching pursuit Matching pursuit is a commonly used algorithm for sparse representation [18], and the principle of matching pursuit lies in decomposing a signal x into the linear combination of basis functions dγ (‖dγ ‖ ¼ 1) that belong to a redundant dictionary D A Rnq . After greedy search, an atom dγ 0 A D that best matches the signal structure is selected, and the signal x can be decomposed into (1) x ¼ x; dγ 0 dγ 0 þ R1 x where hi means inner product, R1 x is the residue after the first matching. For the orthogonality of dγ 0 and R1 x, we get ‖x‖2 ¼ x; dγ 0 2 þ‖R1 x‖2 (2)
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where ‖‖ means 2-norm. The residue R1 x is further decomposed in the same way as x. After applying N times similar procedure to the residue, dγ n is selected if Rn x; dγ n is the maximum over the dictionary, then x¼
N 1 X
Rn x; dγ n dγ n þ RN x
(3)
n¼0
This procedure is repeated until a desired approximated precision is reached (the final residue RN x is small enough). The matching pursuit approach works well only when the dictionary has the similar structure with the signal. In practice, however, it is difficult to find a robust parametric basis function that achieves goals of the feature extraction, which restricts the application of matching pursuit seriously.
3. The extraction of defect-induced impulse based on sparse decomposition 3.1. Construct the sparse dictionary based on correlation filtering When a localized fault occurs in rolling element bearing, the fault impact will produce structure resonance and generate impulse response signal with an exponential damping amplitude [19]. Consequently, damped second-order mass-springdamper system and its unit-impulse response function can be used to describe the vibration response of bearing fault, and the basis function of the sparse dictionary can be defined as 2 3 6 2πζ 7 dγ ðtÞ ¼ exp4qffiffiffiffiffiffiffiffiffiffiffiffiffif d ðt t 0 Þ5 sin 2π f d ðt t 0 Þ ; t Z t 0 (4) 2 1ζ where f d is the natural frequency of the bearing system. ζ represents the relative damping ratio which is generally less than 0.2 for steal structure [20]. t 0 is the moment that impact occurred. Obviously, the parameters ðf d ; ζ ; t 0 Þ, which decide the waveform of atoms, have an influence on the result of sparse decomposition. In the following, the parameters ðf d ; ζ ; t 0 Þ are optimized by correlation filter method [11] to approximate those impulses in the bearing signal xðtÞ. The followings are the procedure: (1) Let f d A f lc : Δf d : f s =2 , ζ A 0:001: Δζ : 0:2 and t 0 A 0: Δt: T c , where Δfd , Δζ and Δt are the step-sizes of ðf d ; ζ ; t 0 Þ, f lc is the lower cutoff frequency, f s is sampling frequency, and T c is the signal duration that used for correlation filtering. The initial dictionary C is generated according to Eq. (4). For the periodic characteristic of the impulse response, only a short time duration T c which includes several impacts is needed. (2) Calculate the correlation coefficients λðtÞ between the bearing signal xðtÞ and the basis atoms dγ in the initial dictionary C. xðtÞ; dγ ðtÞ λðtÞ ¼ (5) ‖xðtÞ‖2 ‖dγ ðtÞ‖2 (3) Search all the local maximums λi ði ¼ 1; …NÞ from λðtÞ, and the corresponding natural frequencies, relative damping ratios and the occurrence times of impulses are denoted as ðf di ; ζ i ; t 0i Þ, i ¼ 1; …N. The search precision of above three parameters is directly influenced by the step-sizes Δfd, Δζ, Δt. (In practice, bearing fault may evoke several degrees of natural frequencies, meanwhile, considering the quantization error, the identified f di and ζ i of different local maximums λi may be same or close.) (4) Merge the identified natural frequencies close to each other into their middle value. In this paper, the area is set to f di 2Δf d ; f di þ 2Δf d . Let f dj ðj ¼ 1; …JÞ be the natural frequencies after merged, and ζ j be the corresponding relative damping ratios. (5) Put all the f dj ; ζ j ðj ¼ 1; …JÞ and t 0i ði ¼ 1; :::NÞ into Eq. (4), and get the optimized dictionary D with J N atoms. Obviously the redundancy of dictionary D is far less than that of initial dictionary C, which can largely improve the speed of sparse decomposition. Since the parameters of dictionary D are determined from the bearing signal xðtÞ, all the parameters have clear physical meaning. Moreover, several natural frequencies and damping ratios are used for constructing the impact response, which is in conformity with practical situations. 3.2. The proposed fault feature extraction method based on sparse decomposition To improve the calculation speed of sparse coefficients solving and make sparse representation more suitable for bearing fault diagnosis, the proposed method employs correlation filtering to get the optimized dictionary parameters based on the bearing signal firstly. Then the bearing signal is divided into several segments based on the smallest fault characteristic period, and the matching pursuit is carried out by segments. Moreover, to improve the extracting accuracy of the impact
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Fig. 1. Flowchart of the proposed fault feature extraction algorithm for rolling element bearing.
moment, the parameters of t 0i in the dictionary is further refined when executing matching pursuit. The process of the proposed algorithm is shown in Fig. 1. The details of this algorithm are summarized as follows: (1) Collect the bearing signal xðtÞ by acceleration transducer. (2) High pass filtering is carried on to remove the low frequency components caused by manufacturing and assembling errors and other structure vibration. Let xp ðtÞ be the signal after filtering. (3) Repeat the steps of Section 3.1, the dictionary parameters f dj ; ζ j ðj ¼ 1; …JÞ and t 0i ði ¼ 1; …NÞ are obtained from the signal xp ðtÞ by correlation filtering. (4) Let T s be the smallest appearance period of possible defect-induced impulse (which is corresponding to the inner race fault characteristic period). Divide the signal xp ðtÞ into W segments xw ðtÞðw ¼ 1; 2; ⋯; WÞ with each segment having a length of T s . For the single fault situation, this step can avoid more than one impact happened during one period. (5) Matching pursuit is applied for every segment xw ðtÞ to extract the impulse signal xws . Since we divide signal into segments by the characteristic period of inner race fault, there may be no impulse in some segments when local defect appears in other bearing parts. Even for inner race fault, the defect-induced impulse might be too weak to be detected when the defect location rotates out of the bearing load zone. In view of above situation, we utilize the parameter t 0i obtained from the correlation filtering to prevent the mismatch problem, and the details are as follows: A) Perform the division operation ðt 0i =T s Þ and seek its quotient and remainder, that is, ki ¼ roundðt 0i =T s Þ, τi ¼ modðt 0i ; T s Þ. B) Compare ki with w, when w aki , we can assume that there is no impulse or the impulse is too weak to be detected in the segment xw ðtÞ. In this case, the matching pursuit is not needed, and the reconstructed signal of this segment is set to zero. C) For segment xw ðtÞ that w ¼ ki , extend the remainder τi by the step-size Δτ to 2M þ 1 points, that is,τ0i ¼ ½ M Δτ þ τi ; M Δτ þ τi . Get the optimized dictionary Di by substituting f dj ,ζ j ðj ¼ 1; …JÞ and τ0i into the Eq. (4), and the atoms number of Di is J ð2M þ1Þ. Matching pursuit is applied to segment xw ðtÞ until the iteration v termination condition is reached. Let the best matching atom be dγ and corresponding matching coefficient cwv ¼
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D E v Rwv xw ; dγ ðv ¼ 0; :::V 1Þ, which are obtained at the end of the sparse decomposition. The reconstructed signal xws is xws ¼
VX 1 v¼0
v
cwv dγ
(6)
(6) The W segments of the reconstructed signals are sequentially connected and the impulse component of the bearing signal is obtained by xs ðtÞ ¼
W X
xws ½t þ ðw 1ÞT s
(7)
w¼1
(7) Get all the time interval of impulses in xs ðtÞ and take the average after discarding the value that changes largely, and obtain the fault feature finally.
4. Simulation analysis Considering the fact that several degrees of natural frequencies may be evoked by a local fault, the single point defect signal of rolling element bearing x(t) can be described through extending the bearing model proposed by Mcfadden [19] to
Fig. 2. Simulation signals of outer race fault with no noise. (a) Signal without slipping. (b) Signal with large slipping. (c) Amplitude spectrum of (a). (d) Amplitude spectrum of (b). (e) Hilbert envelop spectrum of (a). (f) Hilbert envelop spectrum of (b).
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Fig. 3. The simulated noisy signals (red line represents pure signal; blue line represents noisy signal). (a) SNR ¼ 0 dB. (b) SNR¼ 5 dB. (c) SNR¼ 10 dB. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 4. The process of correlation filtering for the pure signal with no slipping.
the following form 2
3 h i 6 2πζ j 7 Aij exp4qffiffiffiffiffiffiffiffiffiffiffiffiffif dj ðt τi iT Þ5 sin 2π f dj ðt τi iT Þ þ ηðt Þ x ðt Þ ¼ 2 i¼1j¼1 1 ζj J I X X
t Z τi
(8)
where ðf dj ; ζ j Þ j ¼ 1; …; J represent the jth natural frequency and corresponding relative damping ratio. Aij is the amplitude of the ith transient response under the jth natural frequency. T is the time period of the defect-related impulses, τi is the delay which is a random variable for the time lag between impacts due to the presence of sliding. ηðt Þ is additive background noise. Obviously, τi ¼ 0 when there is no sliding between the rolling elements.
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Fig. 5. (a) The identified results of natural frequencies and relative damping ratios for the pure signal with no sliding. (b) The identified natural frequencies for the case of large sliding. (c) The identified relative damping ratios for the case of large sliding.
Let the simulated bearing has the same parameters as bearing NUP311EN (pitch diameter D ¼ 85 mm, ball diameter d ¼ 18 mm, ball number z ¼ 13, contact angle α ¼ 0o ). Let f n be the shaft rotational frequency, and the ball passing inner race frequency (BPFI) f I and ball passing outer race frequency (BPFO) f O can be calculated by [9]
z fn d 1 cos α (9) fO ¼ D 2 fI ¼
z fn d 1 þ cos α D 2
(10)
When f n ¼ 10 Hz, the characteristic defect frequencies of bearing NUP311EN are: f O ¼ 51:24 Hz and f I ¼ 78:76 Hz. Let J ¼ 2, f d1 ¼ 1800 Hz, ζ 1 ¼ 0:03, f d2 ¼ 4200 Hz, ζ 2 ¼ 0:01, Aij is arbitrary constant distributed in the range of 0–10. Let the sampling frequency f s be 214 ¼16384 Hz, and sampling time is set to be 0.5 s. Let T ¼ 1=f O ¼ 19:52 ms in Eq. (8) for simulating outer race fault. Let τi ¼ 0 to simulate the case without sliding of rolling element and τi ¼ 0:5T randð0; 1Þ to simulate large sliding of rolling element, and the simulation signals are given in Fig. 2. By comparing Fig. 2(a) with (b), it can be seen that because of the influence of rolling element sliding, the time intervals between any two impulses are no longer fixed value, and serious frequency aliasing happens in Fig. 2(d). The two signals are both demodulated in the same bandwidth [1600, 2000] Hz and the envelop spectrums are given in Fig. 2(e) and (f), respectively. It can be seen that the fault characteristic frequency f O and its harmonics are no longer obvious in the demodulation spectrum for the large sliding of rolling element.
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Fig. 6. The reconstructed impulses under different SNR (red lines: reconstructed signal, blue line: theoretical signal). (a) Pure signal. (b) SNR ¼0 dB. (c) SNR ¼ 5 dB. (d) SNR¼ 10 dB. Table 1 Impulse interval of the reconstructed signals under different SNRs.
Average (ms) Relative error
set vale
pure signal
0 dB
5 dB
10 dB
19.52 /
19.45 0.36%
19.45 0.36%
19.43 0.46%
19.15 1.90%
A white gaussian noise is added to the simulated signal with large rolling element sliding. Let the signal to noise ratio (SNR) be 0 dB, 5 dB and 10 dB, respectively, and the noisy signals are given in Fig. 3. As can be seen, with the increasement of noise level, the fault impacts are buried in the noise gradually and become hard to detect at last. The above simulation signals are decomposed and reconstructed by the proposed fault feature extraction method. When searching the parameters by correlation filtering, the search scopes of f d ,ζ and t 0 are set as: f d A ½0: 10: f s =2, ζ A ½0:001: 0:001: 0:2 and t 0 ¼ ½0: T=24 : 0:5, respectively. The process of correlation filtering for the case τi ¼ 0 and ηðtÞ ¼ 0 is given in Fig. 4. The identified natural frequencies and corresponding relative damping ratios in different SNRs are given in Fig. 5 (where the identified natural frequencies are sorted from small to large before displaying). As is shown Fig. 5, both the two parameters are accurately identified when τi ¼ 0 and ηðtÞ ¼ 0, the identified accuracy of the natural frequency f di still keep high even under large sliding of rolling element and low SNR, and although the identified accuracy of relative damping ratios ζ i is degraded with the increasement of noise level, most the identified results distribute near the theoretical values. Considering the fact that only the most matching atoms are selected for constructing signal in matching pursuit, a good matching result can be obtained even when only one set of identified atom parameters is close to the real values. Consequently, all the identified parameters are used to construct the redundant dictionary. After dividing the simulation signal xðtÞ by the smallest appearance period T s of possible defect-induced impact (here T s ¼ 1=f I ¼ 1=78:76 ¼ 12:7 ms), matching pursuit is carried out on every segment xw ðtÞ following the step (5) in Section 3.2, where step-size Δτ is set as T I =27 and M is set as 10 when constructing the dictionary Di. The defect-induced impulses are reconstructed and the results are presented in Fig. 6. It can be seen that the reconstructed results match well with the theoretical results although there is a large rolling element sliding. And as shown in Fig. 6(b), the matching accuracy remains good when SNR ¼0 dB, but when SNR decreases to 10 dB, as shown in Fig. 6(d), some undesired signals are wrongly extracted. After discarding the suddenly changed value, the average and relative error of the impulse intervals are given in Table 1. It could be concluded that although the matching accuracy degrades with the decrease of SNR, the average of impact intervals under low SNR is still close to the set value. The simulation results show that the proposed method can extract the bearing fault feature even under large rolling element sliding and low SNR conditions.
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Fig. 7. Experimental facility: (a) The test bench; (b) The artificial fault in outer race.
Fig. 8. The vibration signal picked up from the test bench. (a) Time-domain wave. (b) Amplitude spectrum.
Fig. 9. The signal after high pass filtering xp(t).
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Fig. 10. The identified natural frequencies and relative damping ratios by correlation filtering. Table 2 The natural frequencies and relative damping ratios for optimized dictionary. j
1
2
3
4
5
6
7
8
f dj =ðHzÞ
7550
8000
9350
14750
15100
16150
16900
35750
ζj
0.021
0.026
0.091
0.055
0.050
0.026
0.020
0.011
Fig. 11. The reconstructed impulses for bearing with outer race fault.
Fig. 12. Hilbert envelop spectrum of the reconstructed bearing signal.
5. Experimental analysis 5.1. outer race fault experiment The experiment is conducted on a test bench shown in Fig. 7, the shaft of the test rig is supported by two bearings model N205M (pitch diameter D ¼38 mm, ball diameter d¼6.5 mm, ball number z¼13, contact angle α ¼ 0o ) and driven by an AC motor and a belt pulley. An artificial fault (width 0.2 mm and depth 0.5 mm) is produced on the outer-race of one bearing, and the vibration signal is picked up from the bearing pedestal with sample frequency fs ¼100 kHz. Let shaft rotational speed be 800 rpm, the characteristic defect frequencies (periods) of outer and inner races are fO ¼71.84 Hz (13.92ms) and fI ¼101.5 Hz (9.85ms), respectively. The time domain wave and frequency spectrum of the measured acceleration signal is presented in Fig. 8. The fault signal x(t) is decomposed and reconstructed according to the flowchart presented in Fig. 1. Let the lower cutoff frequency f lc ¼ 1=20f s to remove the low frequency disturbance caused by motor and pulley drive, and the filtered signal xp ðtÞ is shown in Fig. (9). The filtered signal is divided into equal segments based on the smallest fault characteristic period T s (9.85ms). Let f d A ½f lc : 50: f s =2, ζ A ½0:001: 0:001: 0:2 and t 0 ¼ ½0: T=24 : 1, when correlation filtering is carried out on the filtered signal xp ðtÞ. The identified natural frequencies f di and corresponding relative damping ratios ζ i are given in Fig. 10. After merging the adjacent values of f di , the parameters of optimized dictionary are shown in Table 2, and the final reconstructed signal is given in Fig. 11. By comparing Fig. 9 with Fig. 11, it can be seen that most interference components are removed in the reconstructed signal. After discarding the intervals that change largely, the average of the impact intervals in the reconstructed signal is 14.00 ms, which is very close to the characteristic defect period of outer race (13.92 ms). The
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Fig. 13. The transmission test bench and the fault bearing. (a) The tested transmission. (b) The position of sensor. (c) The artificial fault in inner race.
Fig. 14. The vibration signal picked up from the transmission. (a) Time-domain wave. (b) Amplitude spectrum.
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Fig. 15. The gearbox signal after high pass filtering xp(t).
Fig. 16. The identified results of natural frequencies and relative damping ratios of the gearbox signal. Table 3 The parameters of optimized dictionary. j
1
2
3
4
5
6
7
8
f dj = ðHzÞ
5450
6000
7650
12200
16200
17050
17850
45750
ζj
0.039
0.044
0.031
0.016
0.004
0.006
0.003
0.006
Fig. 17. The reconstructed impulses of the gearbox signal.
reconstructed signal is further demodulated in full frequency band, and the Hilbert envelope spectrum is shown in Fig. 12, in which the characteristic defect frequency of outer race f O and its harmonics are clearly exposed. Therefore, the outer race defect is identified. 5.2. Inner-race fault experiment To further verify the effectiveness of the proposed method, experiment is conducted on a five-speed automobile transmission. An artificial fault (width 0.2 mm and depth 1 mm) is produced on the inner-race of the bearing NUP311EN which supports the output shaft. The transmission test bench and the fault bearing are shown in Fig. 13. The rotational speed of the output shaft is set as 500 rpm, the characteristic defect frequencies (periods) of outer and inner races are computed by Eqs. (9) and (10), and the values are fO ¼42.75 Hz (23.39 ms) and fI ¼65.53 Hz (15.22 ms), respectively. Let the sample frequency fs ¼ 100 kHz, the vibration signal picked up from the bearing pedestal of output shaft is shown in Fig. 14. It can be seen that there is no visible impact in the time domain wave, and the amplitude spectrum is dominated by constantly mesh frequency and its harmonics. The signal is processed by the proposed fault feature extraction method with the same process parameters as Section 5.1. The signal after filtering is given in Fig. 15, where no clear fault impulse can be observed. Again, after the correlation filtering, the dictionary parameters are identified and the results are given in Fig. 16. By utilizing the optimized natural frequencies and relative damping ratios shown in Table 3, the fault impulses are reconstructed by segments and the final results are given in Fig. 17. As can be seen in Fig. 17, the periodicity of the impulses experiences a certain degree of randomness, and the cyclic statistics characteristics of the bearing are revealed clearly. This
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Fig. 18. Hilbert envelop spectrum of the reconstructed gearbox signal.
phenomenon is caused by the slippage as the rolling elements enter and leave the bearing load zone. After discarding the intervals that change largely, the average of the impact intervals is 15.47 ms. Comparing with the characteristic defect periods of bearing NUP311EN, there is only 1.6% relative error with the characteristics defect period of inner race (15.22 ms). Similarly, the reconstructed signal is further demodulated in full frequency band, and the Hilbert envelope spectrum is shown in Fig. 18. As can be seen, not only the characteristic defect frequency of inner race f I and its harmonics but also their sidebands are clearly detected. Thus we can confirm that fault should locate in the inner race of the bearing NUP311EN. The effectiveness of the proposed method is further verified.
6. Conclusions This paper proposes a bearing fault feature extraction method based on sparse representation. According to the characteristic of the defect-induced impulse, sparse dictionary is constructed by unit-impulse response function of damped second-order system, and the natural frequencies and relative damping ratios are obtained directly from bearing signal by correlation filtering method. Considering the influences of speed fluctuation, rolling element sliding and other time-varying factors, the defect induced impulses are reconstructed by segments. The results of both simulated and actual bearing vibration signals show that the proposed method is effective to extract the rolling bearing fault impulses even under large rolling element sliding and low signal to noise ratio.
Acknowledgments This work was supported by the National Natural Science Foundation of China (Nos. 51475169 and 51475170) and the Natural Science Foundation of Guangdong Province (S2013030013355).
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